Density, Upthrust & Viscous Drag (Edexcel International A Level (IAL) Physics): Structured Questions

A hollow brass cylinder with closed ends is floating on the surface of water.

The cylinder has a length of 4.0 cm and an external diameter of 2.1 cm as shown.

63% of the volume of the cylinder is submerged. The cylinder contains negligible weight of air.

Explain why the brass cylinder floats.

The density of water is 1.0 × 10³ kg m⁻³

i) Show that the mass of the cylinder is about 9 × 10⁻³ kg.

(4)

ii) Deduce whether an identical hollow cylinder made of gold would also float.  

Assume that the volume of gold is the same as the volume of brass.  

density of gold = 19.3 × 10³ kg m⁻³

density of brass = 8.7 × 10³ kg m⁻³

(4)

A student carried out an experiment to determine the viscosity of a liquid. He measured the terminal velocities v of several different glass spheres of diameter d, as they fell through the liquid.

The student used his measurements to plot the graph of v against d² shown below.

Explain what is meant by terminal velocity.

A glass sphere of diameter 35 mm is travelling through the fluid at terminal velocity.

Show that the drag force on this sphere is about 0.2 N.

density of glass ρ_g = 2.52 × 10³ kg m⁻³

density of liquid ρ_s = 1.43 × 10³ kg m⁻³

The student reads in a textbook that if Stokes’ law is obeyed

            $v=k{d}^2$

where k is a constant.

i) Deduce from the graph whether the flow of liquid around the spheres was laminar.

(3)

ii) Determine a value for k using the student’s graph.

k = .............................(3)

iii) The constant k is given by

$k=\frac{\left({\rho }_g-{\rho }_s\right)g}{18\eta }$

where η is the viscosity of the liquid.

Determine a value for η.

density of glass ρ_g = 2.52 × 10³ kg m⁻³

density of liquid ρ_s = 1.43 × 10³ kg m⁻³

η = ...........................(2)

A small sphere is moving horizontally through a viscous liquid.

Stokes' law can be used to calculate the drag force on an object.

State the conditions that must apply for Stokes' law to be valid.

There is a constant force of 2.3 × 10⁻⁵ N acting horizontally on the sphere.

diameter of sphere = 4.5 × 10⁻³ m    

viscosity of liquid = 7.1 × 10⁻² Pa s

i) At one instant, the speed of the sphere is 5.2 × 10⁻³ m s⁻¹.  

Calculate the resultant horizontal force on the sphere.

Resultant horizontal force = .................................(3)

ii) Calculate the maximum speed of the sphere in the horizontal direction.

Maximum horizontal speed = .............................(2)

A larger diameter sphere in the same liquid is acted upon by the same constant force as in (b). The liquid is at a lower temperature. 

Explain the effect these changes have on the maximum speed of this sphere.

A student investigated the viscosity of olive oil at different temperatures using a steel sphere of radius $r$.

Describe how the student can make measurements of the steel sphere to determine its radius accurately.

Explain how the student can ensure that Stokes' law will apply to the steel sphere as it falls through the olive oil.

The student dropped the steel sphere into a tall cylinder of olive oil, as shown, and timed the sphere as it fell through the oil at terminal velocity.

The student placed two rubber bands around the cylinder and started a stopwatch when the sphere passed the first band and stopped it when the sphere passed the second band.

The student recorded the time $t$ for the ball to fall through a distance of $21.0\text{ cm}$ and repeated this for three different temperatures. They calculated the viscosity $\eta$ at each temperature using the formula

$\eta =\frac{2r^{2}g\left({\rho }_s-{\rho }_o\right)}{9v}$

radius of steel sphere $r=2.2\text{ mm}$
density of steel ${\rho }_s=7800{\text{ kg m}}^{-3}$
density of olive oil at 20 °C ${\rho }_o=920{\text{ kg m}}^{-3}$

Their results are shown in the table.

The student concluded that the viscosity of olive oil decreases linearly with temperature.

Criticise the student's investigation and conclusion.

A student determines the viscosity of an engine oil at room temperature using the falling-ball method.

The student releases a small steel ball from rest at the surface of the oil in a tall tube, as shown.

Sketch a graph to show how the velocity of the ball varies with time, from the instant of release until after the ball reaches terminal velocity.

Describe how the student obtains an accurate value for the terminal velocity of the ball.

Explain why the temperature of the oil must be kept constant during the experiment.

The ball has a diameter of $3.0\text{mm}$. The ball falls a distance of $0.240\text{m}$ in a time of $1.6\text{s}$ while travelling at terminal velocity.

density of steel $=7800{\text{kg m}}^{-3}$

density of the oil $=900{\text{kg m}}^{-3}$

Calculate the viscosity of the oil.